Capillary pressure curves are widely used in material, soil and environmental sciences, and especially in the petroleum industry. Capillary pressure curves provide critical information frequently used in the assessment of the economic viability of oil reservoir development.
Capillary pressure may be obtained by either mercury intrusion, porous plate, or centrifuge methods. The mercury intrusion method is rapid, but it is destructive, and the mercury/vacuum system does not represent the wettability of reservoir system. The porous plate method is a direct and accurate technique, but is extremely time-consuming, since the equilibrium time can range from a week to months per pressure point.
The centrifugal capillary pressure curve technique was introduced by Hassler and Brunner in 1945, as described in Hassler, G. L., Brunner, E., “Measurement of Capillary Pressure in Small Core Samples”, Trans. AIME, 1945, 160, 114-123 and N. T. Burdine, Trans. AIME 198, 71 (1953) which is incorporated herein by reference. This technique, which involves rotating fluid bearing rock cores at variable speeds in a specially modified centrifuge, has been extensively investigated, and is commonly used in the petroleum industry. Sample rotation yields a centrifugal force which will empty pores with matching capillary forces. Collecting the expelled fluid as a function of increasing rotational speed permits a quantification of the capillary pressure as a function of fluid content or saturation.
The Hassler-Brunner centrifugal capillary pressure technique, which involves rotating fluid bearing rock cores at variable speeds in a specially modified centrifuge, has been extensively investigated, and is commonly used in the petroleum industry. Sample rotation yields a centrifugal force which will empty pores with matching capillary forces. Collecting the expelled fluid as a function of increasing rotational speed permits a quantification of the capillary pressure as a function of fluid content or saturation.
The conventional interpretation of centrifugal capillary pressure data is based on several assumptions: (1) nonlinearity of the centrifugal field is not significant; (2) gravity has no effect on fluid distribution; and (3) the capillary pressure is zero at the bottom (outlet end-face) of the core plug. These assumptions are known to lead to significant errors in the measurement of the capillary pressure curve. In addition, these three conditions can not be simultaneously satisfied. The first assumption requires a short sample and large rotational radius. For low capillary pressures, the experiment requires a very low rotational speed. In this case, the effect of gravity can not be neglected. For high capillary pressures, the experiment requires a very high rotation speed, which is likely to lead to a violation of the third assumption (capillary pressure is zero at the outlet). In addition, the rock pore structure in unconsolidated or friable samples (for example marginal reservoirs) will change due to the high centrifugal forces, thereby altering the capillary pressure curve.
Conventional centrifuge methods for capillary pressure determination are time consuming and special instrumentation is required for the experiment. Measurement of the full capillary pressure curve requires approximately 15 different centrifuge speeds, thus requiring one day to several days for measurement. In addition, some friable and unconsolidated rock samples may be broken during ultracentrifugation, as described in D. Ruth and Z. Chen, The Log Analyst 36, 21 (1995). The experiment requires a very expensive ultracentrifuge with precise speed control over a wide range of speeds. A special core holder and stroboscope for collecting and measuring expelled liquid are also necessary for the experiment.
Magnetic Resonance Imaging (MRI) can also be used for capillary pressure determination. NMR is a powerful, non-destructive, measurement method, which, with techniques developed by the inventors described in Balcom, B. J., MacGregor, R. P., Beyea, S. D., Green, D. P., Armstrong, R. L. and Bremner, T. W. “Single Point Ramped Imaging with T1 Enhancement (SPRITE)”, J. Magn. Res. A (1996) 123, 131-134, offer unique advantages in the measurement of spatially resolved fluid saturation in porous media, discussed in Chen, Q., Gingras, M. and Balcom, B. J., “A magnetic resonance study of pore filling processes during spontaneous imbibition in Berea sandstone”, J. Chem. Phys., 119, 9609-9616 (2003) and Balcom, B. J., Barrita, J. C., Choi, C., Beyea, S. D., Goodyear, D. J. and Bremner, T. W. “Single-point magnetic resonance imaging (MRI) of cement based materials”, Materials and Structures (2003) 36, 166-182.
Capillary pressure measurements can be obtained through the use of a centrifuge and MRI equipment as described in U.S. Pat. No. 7,352,179 which is incorporated herein by reference (hereinafter referred to as “GIT-CAP”). To fully measure the capillary pressure curve, several centrifuge speeds (typically 3 to 5) are required.
For example, FIG. 1 illustrates a capillary pressure curve acquired with measures obtained using 5 centrifuge speeds. The solid line in the plot of FIG. 1 is the conventionally (centrifuge only) acquired capillary pressure curve. The data points in the different centrifuge speeds Pc cover different portions of the Pc curve and can be acquired according to one or more of the GIT-CAP methods. One set of data points from a single speed alone does not fully define the Pc curve but it is enough information to fulfill a Pc model that would fully define the curve.
Another saturation dependant measurement is relative permeability. Relative permeability describes the flow of one fluid or gas at different saturation levels of another fluid or gas. The flow rate at 0% level of the other fluid/gas is 1.0 by definition. One method of obtaining relative permeability using MRI equipment is described in U.S. patent application Ser. No. 11/808,300 filed on Jun. 8, 2007, which is incorporated herein by reference.
Conventional Capillary Pressure Models
Capillary pressure models exist for modelling capillary pressure curves using capillary pressure measurements obtained from conventional methods (other than MRI). The two most popular are the Brooks-Corey (Brooks, R H, Corey, A T, “Hydraulic properties of porous media” Hydrol. Pap., 3, Colo. State Univ., Fort Collins (1964)) and the van Genuchten (van Genuchten, M. T. “A Closed-form equation for predicting the hydraulic conductivity of unsaturated soils”, Soil Sci. Soc. Am. J., 44, 892-298 (1990)) models which are incorporated herein by reference. The Brooks-Corey model uses the residual water saturation and therefore in cases where it is known may provide a better fit. After selecting a model, the least squares fit is applied to fit the model parameters.
Brook-Corey Pc Model
The Brooks-Corey model relates capillary pressure and water saturation as:
                              P          C                =                                            p              e                        ⁡                          (                              S                W                *                            )                                            -                          1              λ                                                          (        1        )            
Where pe and λ are fitted parameters, PC is the normalized wetting-phase (water for air water) saturation and is defined as follows:
                              S          w          *                =                                            S              w                        -                          S              wr                                            1            -                          S              wr                                                          (        2        )            
Where Sw is the water saturation and Swr is the residual water saturation. Substituting (2) in (1) and rearranging for Sw we get:
                              S          W                =                                            (                              1                -                                  S                  wr                                            )                        ⁢                                          (                                                      P                    C                                                        P                    e                                                  )                                            -                λ                                              +                      S            wr                                              (        3        )            van Genuchten Pc Model
The van Genuchten model related Pc and water saturation as:
                              P          C                =                              1            α                    ⁢                                    (                                                S                  w                                      -                                          1                      M                                                                      -                1                            )                                      1              N                                                          (        4        )            
Where PC is the capillary pressure, Sw is the water saturation and α, M and N are fit parameters where N and M are related by:
                    N        =                  1                      1            -            M                                              (        5        )            
Substituting equation (5) into (4) and rearranging for Sw gives:
                              S          w                =                              (                          1              +                                                (                                      α                    ⁢                                                                                  ⁢                                          P                      C                                                        )                                N                                      )                                -                                          N                -                1                            N                                                          (        6        )            Least Fit Error
There a many ways to calculate the model error and then minimize that error to find the optimal solution (fit). The simplest error is the summation of the difference between the measured Y value and the calculated (modelled) Y value squared as described in:Error=Σ(Sw-measured−Sw-calculated)2 
This will place equal weighting on all Y values and assumes the X values have little or no error. This works well for linear data (i.e. y=mx+b). For capillary pressure curve data the data is not linear and is exponential in nature. A better error equation is the percentage change in the Y values.
                    Error        =                  ∑                                    (                                                                    S                                          w                      -                      measured                                                        -                                      S                                          w                      -                      calculated                                                                                        S                                      w                    -                    measured                                                              )                        2                                              (        8        )            
This places a 1/X weighting on the error putting more importance on the Y values that are closer to X=0 (lower saturations). This produces a better fit for Pc data. The error in saturation units is lower at lower saturation values.